reserve x,y,z for object, X,Y for set,
  i,k,n for Nat,
  p,q,r,s for FinSequence,
  w for FinSequence of NAT,
  f for Function;

theorem Th37:
  for f being DTree-yielding Function holds dom doms f = dom f &
  doms f is Tree-yielding
proof
  let f be DTree-yielding Function;
A1: dom doms f = dom f by FUNCT_6:def 2;
  hence dom doms f c= dom f;
  thus dom f c= dom doms f by A1;
  now
    let x;
    assume x in dom doms f;
    then
A2: x in dom f by A1;
    then reconsider g = f.x as DecoratedTree by Th24;
    (doms f).x = dom g by A2,FUNCT_6:22;
    hence (doms f).x is Tree;
  end;
  hence thesis by Th22;
end;
